Formulas for numbers of molecules on the surface of a large nucleus.

by Leonid Sakharov

Numerical simulation of nucleation phenomena during crystallization demands a definition several parameters of the clusters of crystalline molecules in initial amorphous phase. Among of such key parameters are a numbers of molecules on the surface of cluster these can be transformed from one phase to other changing number of molecules in the cluster.

Let’s cluster has i molecules with volume v_m each.

Effective size of d is defined by equation v_m = d³

The volume of the cluster is:

V[i] = (4π/3) * r³ = i * v_m

where r is effective radius of the cluster.

The volumes of surface areas where molecules can move from crystal into feeding environment V_s- or into cluster V_s-  from surrounding the nucleus phase are defined by formulas:

V_s- = (4π/3) * (r³ – (r – g*d)³)

V_s+ = (4π/3) * ((r + g*d)³ - r³)

where g is geometrical coefficient that has meaning of effective thickness of surface layer measured in size of molecule. Let also introduce a constant p = 4π/3 = 4.188790205 that will help to simplify an equations.

Number of molecules on surface of the cluster can be calculated by expressions:

N_s- = V_s-/v_m

N_s+ = V_s+/v_m

Following are series of mathematical transformations to present N_s- and N_s+  in most convenient for calculation form:

N_s- = (p/v_m) * ( r³ – r³ + 3 r²*gd – 3 r*g²d² +g³d³)

N_s+ = (p/v_m) * (r³ + 3 r²*gd + 3 r*g²d² +g³d³  - r³)

Defining radius of nucleus from number of molecules:

r = p^-1/3 * i^1/3*v_m^1/3

and substituting it in equation for number molecules on surface of nucleus the following formula appears:

N_s- = (p/v_m) * (3 p^-2/3 * i^2/3*v_m^2/3*g*v_m^1/3 – 3 p^-1/3 * i^1/3*v_m^2/3*g² v_m^2/3  +g³ v_m)

N_s+ = (p/v_m) * (3 p^-2/3 * i^2/3*v_m^2/3*g*v_m^1/3 + 3 p^-1/3 * i^1/3*v_m^2/3*g² v_m^2/3  +g³ v_m)

And finally getting equations:

N_s- = 3 p^1/3 * i^2/3*g – 3 p^2/3 * i^1/3*g²  +g³

N_s+ = 3 p^1/3 * i^2/3*g + 3 p^2/3 * i^1/3*g²  +g³

Taking that the value of geometrical coefficient could be safely estimated having value 0.85 formulas above can be presented as

N_s- = 4.1*i^2/3 – 5.6*i^1/3  + 0.6

N_s+ = 4.1*i^2/3 + 5.6*i^1/3  + 0.6

It is important to point out on the remarkable fact that formulas above do not include a volume of molecule.

Next figure shows numbers of molecules on surface of cluster calculated by formulas above.

There is significant difference between number molecules in the side of cluster and outside it these belong to amorphous phase. This difference is specially is pronounced for small clusters creating pure geometrical advantage for agglomeration of molecules.

Submitted at Aug. 18, 2010; 15:16